NBER WORKING PAPER SERIES

TRACING VALUE-ADDED AND DOUBLE COUNTING IN GROSS EXPORTS

Robert KoopmanZhi Wang

Shang-Jin Wei

Working Paper 18579http://www.nber.org/papers/w18579

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138November 2012

The paper has received research funding from the U.S. International Trade Commission and ColumbiaBusiness School. The views in the paper are solely those of the authors and may not reflect the viewsof the USITC, its Commissioners, the National Bureau of Economic Research, or of any other organizationthat the authors are affiliated with. We are deeply grateful to the editor and the referees for valuablecomments, which have significantly improved the quality and readability of the paper. We also thankPeter Dixon for constructive and generous discussions. We also thank numerous conference participantsand other colleagues especially Zhu Kunfu for valuable comments.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications.

© 2012 by Robert Koopman, Zhi Wang, and Shang-Jin Wei. All rights reserved. Short sections oftext, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit,including © notice, is given to the source.

Tracing Value-added and Double Counting in Gross ExportsRobert Koopman, Zhi Wang, and Shang-Jin WeiNBER Working Paper No. 18579November 2012JEL No. F10

ABSTRACT

This paper proposes a framework for gross exports accounting that breaks up a country’s gross exportsinto various value-added components by source and additional double counted terms. By identifyingwhich parts of the official trade data are double counted and the sources of the double counting, itbridges official trade (in gross value terms) and national accounts statistics (in value added terms).Our parsimonious framework integrates all previous measures of vertical specialization and value-addedtrade in the literature into a unified framework. To illustrate the potential of such a method, we presenta number of applications including re-computing revealed comparative advantages and the magnifyingimpact of multi-stage production on trade costs.

Robert KoopmanResearch DivisionOffice of EconomicsUS International Trade Commission500 E Street SWWashington, DC 20436Robert.Koopman@usitc.gov

Zhi WangResearch DivisionOffice of EconomicsUS International Trade Commission500 E Street SWWashington, DC 20436zhi.wang@usitc.gov

Shang-Jin WeiGraduate School of BusinessColumbia UniversityUris Hall 6193022 BroadwayNew York, NY 10027-6902and NBERshangjin.wei@columbia.edu

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1. Introduction

As different stages of production are now regularly performed in different countries,

intermediate inputs cross borders multiple times. As a result, traditional statistics on trade values

become increasingly less reliable as a gauge of the value contributed by any particular country.

This paper integrates and generalizes the many attempts in the literature at tracing value added

by country and measuring vertical specialization in international trade. We provide a unified

conceptual framework that is more comprehensive than the current literature. By design, this is

an accounting exercise, and does not directly examine the causes and the consequences of global

production chains. However, an accurate and well-defined conceptual framework to account for

value added by source country from available data is a necessary step toward a better

understanding of all these issues.

Supply chains can be described as a system of value-added sources and destinations.

Within a supply chain, each producer purchases inputs and then adds value, which is included in

the cost of the next stage of production. At each stage in the process, as goods cross an

international border, the value-added trade flow is equal to the value added paid to the factors of

production in the exporting country. However, as all official trade statistics are measured in gross

terms, which include both intermediate inputs and final products, they “double count” the value

of intermediate goods that cross international borders more than once. Such a conceptual and

empirical shortcoming of gross trade statistics, as well as their inconsistency with the System of

National Accounts (SNA) accounting standards, has long been recognized by both the economics

profession and policy makers. 1

Case studies on global value chains based on detailed micro data for a single product or a

single sector in industries such as electronics, apparel, and motor vehicles have provided detailed

examples of the discrepancy between gross and value-added trade. According to a commonly

cited study of the Apple iPod (Dedrick, Kraemer, and Linden, 2008), while the Chinese factory

gate price of an assembled iPod is $144, only $4 constitutes Chinese value added. Other case

studies of specific products show similar discrepancies. Case studies, while enhancing our

intuitive understanding of global production chains in particular industries, cannot offer a

comprehensive picture of the gap between value-added and gross trade, and an economy’s

participation in cross-border production chains. Several researchers have examined the issue of 1 See, for example, Leamer et al. (2006), Grossman and Rossi-Hasberg(2008), and Lamy (October 2010).

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vertical specialization on a systematic basis, including the pioneering effort of Hummels, Ishii,

and Yi (2001) (HIY in subsequent discussion). They suggested that a country can participate in

vertical specialization in two ways: (a) uses imported intermediate inputs to produce exports; (b)

exports intermediate goods that are used as inputs by other countries to produce goods for export.

HIY proposed to measure the imported foreign content in a country’s exports based on a

country’s Input-Output (IO) table, which they label as VS (vertical specialization). For a sample

of 11 OECD and 3 non-OECD countries, they calculated that the average share of foreign

content in exports was about 21% in 1990.

There are two key assumptions in HIY's foreign content (VS) estimation: the intensity in

the use of imported inputs is the same between production for exports and production for

domestic sales; and imports are 100% foreign sourced. The first assumption is violated in the

presence of processing exports, which is a significant portion of exports for a large number of

developing countries (Koopman, Wang and Wei, 2008 and 2012). The second assumption does

not hold when there is more than one country exporting intermediate goods.

There is a growing literature in recent years to estimate value-added trade with the advent

of global Inter-Country Input-Output (ICIO) tables based on the Global Trade Analysis Project

(GTAP) and World Input-Output database (WIOD)2, such as Daudin, Rifflart, and Schweisguth

(2011), Johnson and Noguera (2012) and Foster, Stehrer and de Vries (2011). These papers

discuss the connections between their works with HIY, but they are more closely related to the

factor content trade literature. Our paper is the only one in this recent literature to consistently

generalize HIY’s original concepts to a global setting and make HIY a special case of a more

general framework. By integrating the literature on vertical specialization and the literature on

value added trade, this paper expands upon the previous literature in the following five aspects:

First, we provide a unified and transparent mathematical framework to completely

decompose gross exports into its various components, including exports of value added,

domestic value added that returns home, foreign value added, and other additional double

counted terms. Measures of vertical specialization and value-added trade in the existing literature

2 Though usefully global in scope, the GTAP database does not separate imported intermediate and final goods in bilateral trade flows, so improvements have to be made. WIOD is a European Commission sponsored research project to produce better and more up-to-date global ICIO tables, based on a compilation of single-country supply and use tables and detailed bilateral trade statistics for the years 1995-2009. The framework in this paper can be applied to generate a time series decomposition of gross trade flows into their value added and double counted components based on WIOD World IO tables.

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all can be derived from this framework and expressed as some linear combinations of these

components. We show why some of the existing measures are special cases of the generalized

measures in our framework and why some of them have to be modified from their original

definitions in a more general multi-country framework with unrestricted intermediate trade.

Second, rather than simply excluding double counted items from official trade statistics,

we provide an accounting formula that quantifies different types of double counted items for the

first time in the literature. Knowing the structure of the double counted items in a country’s gross

exports can help us to gauge the depth and pattern of that country’s participation in global

production chains. In other words, the relative importance of the various double-counted terms in

addition to value-added trade estimates often contains useful and important information. For

example, in some sectors, China and the United States may have a similar amount of value added

exports. However, the composition of the double counted terms can be very different for the two

countries. For China, the double counted terms may show up primarily in the form of the use of

foreign components (e.g., foreign product designs or machinery) in the final goods that China

exports. For the United States, the double counted terms may show up primarily in the form of

domestic value added finally returned and consumed at home (e.g., product designs by Apple

that is used in the final Apple products produced abroad but sold in the U.S. market). The

structure of these double counted items in addition to their total sums offer additional

information about the U.S. and China’s respective positions in the global value chain.

In addition, we differentiate the double counting terms relative to value-added exports in

a country's gross exports into different types according to whether they should be accounted as

part of a country's GDP and show how they can be quantified, and explain the role they play in

the subtle differences among three related concepts (domestic content in a country’s exports,

value added in exports, and exports of value added) that so far have not always been clearly

distinguished in the literature.

Third, our accounting framework establishes a formal and precise relationship between

value-added measures of trade and official trade statistics, thus providing an observable

benchmark for value-added trade estimates, as well as a workable way for national and

international statistical agencies to remedy the missing information in current official trade

statistics without dramatically changing the existing data collection practices of national customs

authorities.

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Fourth, our estimated global ICIO table may better capture the international source and

use of intermediate goods than in previous databases in two ways. In estimating intermediate

goods in bilateral trade, we use end-use classifications (intermediate or final) of detailed import

statistics rather than the conventional proportionality assumptions. In addition, we estimate

separate input-output coefficients for processing trade in China and Mexico, the two major users

of such regimes in the world. While other studies have used a similar correction for Chinese

exports, the new Mexican IO table provides improved accuracy in estimates of NAFTA trade

flows by distinguishing domestic and Maquiladora production.

Finally, we report a number of applications of our accounting framework and database to

illustrate their potential to reshape our understanding of global trade. For example, with gross

trade data, the business services sector is a revealed comparative advantage sector for India. In

contrast, if one uses our estimated domestic value added (GDP) in exports instead, the same

sector becomes a revealed comparative disadvantage sector for India. The principal reason for

this is how the indirect exports of business services are counted in high-income countries.

Consider Germany. Most of its manufacturing exports embed lots of German domestic business

services. In comparison, most of Indian goods exports use comparatively little Indian business

services. Once indirect exports of domestic business services are taken into account, Indian’s

business service exports become much less impressive relative to Germany and many other

developed countries. As another example, the value added decomposition shows that a

significant portion of China’s trade surplus to the United States in gross trade terms reflects

indirect value added exports that China does on behalf of Japan, Korea and Taiwan. While such

stories have been understood in qualitative terms, our framework offers a way to quantify these

effects.

This paper is organized as follows. Section 2 presents the conceptual framework of gross

exports accounting. Section 3 discusses database construction methods. In particular, we show

how the required inter-country IO model can be estimated from currently available data sources

and report major empirical decomposition results for the year 2004. Section 4 presents a number

of applications that help to illustrate how our gross exports accounting framework may alter our

understanding of issues in international trade and in open-economy macroeconomics. Section 5

concludes.

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2. Gross Exports Accounting: Concepts and Measurement

In this section, we first lay out the main measures of vertical specialization and trade in

value-added as they are originally proposed in the literature. We highlight a key conceptual

difference that separates some measures from others, namely when it is appropriate to include

double-counted items for some purposes but not for others.

We then propose a way to fully decompose a country’s gross exports into the sum of

components that include both the country’s value added exports and various double-counted

components. We further differentiate these double counted terms into different types.

2.1 Concepts

Four measures have been proposed in the vertical specialization and value-added trade

literature:

1. HIY (2001) proposed a measure of vertical specialization from the import side, which

is the imported content in a country’s exports. We follow HIY and label it as VS. It includes both

the direct and indirect imported input content in exports. However, HIY only considered the case

in which the Home country does not export intermediary goods though it imports intermediary

goods from the rest of the world. In mathematical terms, a country's VS in total exports at the

sector level can be expressed as3 :

EAIAVSDM 1)( −−= (1)

2. HIY (2001) also proposed a second measure of vertical specialization from the export

side (which they call VS1). It measures the value of intermediate exports sent indirectly through

third countries to final destinations. However, they did not provide a mathematical definition for

VS1 as they did for VS.

3. Daudin et al (2011) singled out a particular subset of VS1, the value of a country’s

exported goods that are used as imported inputs by the rest of the world to produce final goods

and shipped back to home. They call it VS1*.

4. Johnson and Noguera (2012) defined value-added exports as value-added produced in

source country s and absorbed in destination country r and proposed using value-added to gross

export ratio, the "VAX ratio" as a summary measure of the value-added content of trade.

3 D. Hummels et al. , Journal of International Economics 54 (2001) page 80.

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By definition, as value-added is a "net" concept, double counting is not allowed. As the

first three measures of vertical specialization all involve values that show up in more than one

country’s gross exports, they, by necessity, have to include some double-counted portions of the

official trade statistics. More border crossings by intermediate goods (more double counting)

means a larger difference between trade in value-added and these vertical specialization

measures. This implies that these two type measures are not equal to each other in general

because double counting is only allowed in one of them. They equal each other only in some

special cases as we will show later both analytically and numerically.

In addition, these existing measures are all proposed as stand-alone indicators. No

common mathematical framework proposed in the literature provides a unified accounting for

them and spells out their relationships explicitly. More importantly, as noted earlier, the most

widely used HIY measure (VS) needs two strong assumptions and is only valid in special cases;

there is no mathematically specified measure for indirect exports through third countries, and all

four measures proposed so far do not identify all components in gross exports.

We provide below a unified framework that breaks up a country’s gross exports into the

sum of various well defined components. The value added exports, VS, VS1, and VS1* are

linear combinations of these components. In addition, we show how one may generalize the VS

measure without the restrictive assumption made by HIY (no two-way trade in intermediate

goods). By properly including various double counted terms, our accounting is complete in the

sense that the sum of these well-identified components yields 100% of the gross exports.

For ease of understanding, we start with a discussion of a two-country one sector case in

Section 2.2. We relate the components of our decomposition formula with the existing measures

in the literature in Section 2.3. We provide several numerical examples in Section 2.4 to show

intuitively how our gross exports accounting equation works. Finally, we present the most

general G-country M-sector case.

2.2 Two-country case

Assume a two-country (home and foreign) world, in which each country produces in a single

tradable sector. The good in that sector can be consumed directly or used as an intermediate

input, and each country exports both intermediate and final goods to the other.

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All gross output produced by country r must be used as either an intermediate good or a

final good, either at home or abroad. So country s’s gross output, xs, has to satisfy the following

accounting relationship:

srssrsrssss yyxaxax +++= , r,s = 1,2 (2)

Where ysr is the final demand in country r for the final good produced in Country s, and asr is the

input-output (IO) coefficient, describing units of intermediate goods produced in s used in the

production of one unit gross output in Country r. The two-country production and trade system

can be written as an inter-country input-output (ICIO) model as follows

++

+

=

2221

1211

2

1

2221

1211

2

1

yyyy

xx

aaaa

xx

, (3)

With rearranging, we have

=

++

−−−−

=

−

2

1

2221

1211

2221

12111

2221

1211

2

1

yy

bbbb

yyyy

aIaaaI

xx

. (4)

Since equation (4) takes into account both the direct and indirect use of a country’s gross

output as intermediate goods in the production of its own and foreign final goods, the

coefficients in the B matrix (Leontief inverse) are referred to as “total requirement coefficients”

in the input-output literature. Specifically, b11 is the total amount of Country 1’s gross output

needed to produce an extra unit of the final good in Country 1 (which is for consumption in both

Countries 1 and 2); b12 is the total amount of Country 1’s gross output needed to produce an

extra unit of the final good in Country 2 (again for consumption both at home and abroad).

Similar interpretations can be assigned to the other two coefficients in the B matrix.

We can break up each country’s gross output according to where it is ultimately absorbed

by rearranging both countries' final demand into a matrix format by source and destination, and

rewrite equation (4) as follows:

++++

=

=

2222122121221121

2212121121121111

2221

1211

2221

1211

2221

1211

ybybybybybybybyb

yyyy

bbbb

xxxx

(5)

where ysr is as defined in equation (2), giving the final goods produced in country s and

consumed in country r. This final demand matrix in the middle of equation (5) is a 2 by 2 matrix,

summing along each row of the matrix equals ys, which represents the global use of the final

goods produced in each country as specified in equation (4).

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We label the 2 by 2 matrix on the left hand side of equation (5) the “gross output

decomposition matrix.” It fully decomposes each country’s gross outputs according to where it is

absorbed. Each element xsr in this matrix is the gross output in source country s necessary to

sustain final demand in destination country r. Summing along its row equals total gross output in

country s, xs as specified in equation (2). For example, it breaks up country 1’s gross output x1

into two parts: x1 = x11 + x12. While x11 is the part of Country 1’s gross output that is ultimately

absorbed in country 1, x12 is the part of Country 1’s gross output that is ultimately absorbed in

Country 2.

The RHS of equation (5) further decomposes x11 itself into two parts: x11=b11y11 + b12y21.

The first part, b11y11 is the part of Country 1’s gross output required to produce Country 1’s final

good that is consumed in Country 1. The second part, b12y21, is the part of Country 1’s gross

output that is exported as an intermediate good, and eventually returns home as part of Country

1’s imports from abroad (embedded in foreign final goods).

Similarly, x12 can also be decomposed into two parts: x12 = b11y12 + b12y22. The first part,

b11y12 is the part of Country 1’s gross output that is used to produce exported final good that is

consumed abroad. b12y22 is the part of Country 1’s gross output that is exported as an

intermediate good and used in country 2 to produce final good that is consumed there. Of course,

x1 = x11 + x12 is nothing but Country 1’s total output. By assumption, they are produced by the

same technology and therefore have the same share of domestic value added.

By the same interpretation, Country 2’s gross output also can be first broken up into two

parts: x2 = x21 + x22. x21 is Country 2's gross output that is ultimately absorbed in Country 1,

which can be in turn broken up to b21y11+b22y21. x22 is Country 2’s domestic absorption of its

own gross output, and can be further broken up to b21y12+b22y22.

This conceptual decomposition of a country’s gross output according to where it is

absorbed and further breaking them out in terms of each country's final demand reflect the basic

and uncontroversial Leontief insight, thus is a very useful stepping stone for thinking of a

country’s export of value added.

By the definition of the input-output coefficients, to produce 1 unit of Country 1’s good,

a11 units of domestic intermediate good is used, and a21 units of imported intermediate good is

used. Therefore, the fraction of domestic output that represents the domestic value added in

Country 1 is

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21111 1 aav −−=

Similarly, the share of domestic value added in Country 2’s gross output is:

22122 1 aav −−=

We define V, the 2×2 matrix,

=

2

1

00v

vV . (6)

Multiplying these direct value-added shares with the Leontief inverse B produces the 2×2

value-added share (VB) matrix, our basic measure of value-added shares by source of

production.

=

222212

121111

bvbvbvbv

VB . (7)

Within VB, v1b11 and 222bv denote the domestic value-added share of domestically produced

products for country 1 and country 2 respectively, v2b21 and 121bv denote the share of foreign

country’s value-added in the same goods.4 Because all value added must be either domestic or

foreign, the sum along each column is unity:

1222121212111 =+=+ bvbvbvbv . (8)

Given the assumption on the input-output coefficients, there is no difference in the share

of domestic value added in Country 1’s production for goods absorbed at home versus its

production for exports.5 Therefore, the total domestic value added in Country 1’s gross output is

simply v1x1; it is country 1's GDP by definition.

The total value added in Country 1’s gross outputs can be easily broken up into two parts

based on where it is ultimately absorbed: v1x1 = v1x11 + v1x12, where v1x11 is the domestic value

added that is ultimately absorbed at home, and v1x12 is the domestic value added that is

ultimately absorbed abroad. 4 Note that the VB matrix is not any arbitrary share matrix, but rather the one that reflects the underlying production structure embedded in the ICIO model specified in equations (2) and (3). It contains all the needed information on value-added production by source. 5 Such an assumption is maintained by HIY (2001), Johnson and Noguera (2011), and Daudin et al (2011). One might allow part of the production for exports (processing exports) to take on different input-output coefficients. Such a generalization is pursued by Koopman, Wang, and Wei (2012), who have worked out a generalized formula for computing the share of domestic value added in a country’s gross exports when processing trade is prevalent. However, they have not pursued a total decomposition of a country’s gross exports that allows one to compute the structure of double counted items.

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The last part, v1x12, is also Country 1’s exports of value added. It is instructive to

decompose the last item further. Since v1x12 = v1b11y12 + v1b12y22 , Country 1’s exports of value

added has two components: Country1’s value added embedded in Country 1’s exports of final

good that is absorbed in Country 2 (v1b11y12); and Country 1’s value added in its exports of

intermediate good that is used by Country 2 to produce final good that is ultimately locally

consumed (v1b12y22).

Note that v1x12 is conceptually the same as Country 1’s value added exports as defined by

Johnson and Noguera (2012) except that we express it as the sum of two components related

only to the final demand in the two countries. To summarize, Country 1’s and 2's "value-added

exports" are, respectively:

(9)

Intuitively, there are at least two reasons for a country’s exports of value added to be

smaller than its gross exports to the rest of the world. First, the production for its exports may

contain foreign value added or imported intermediate goods (a). Second, part of the domestic

value added that is exported may return home after being embodied in the imported foreign

goods rather than being absorbed abroad (b). In other words, exports of value added are a net

concept; it has to exclude from the gross exports both foreign value added and the part of

domestic value added that is imported back to home.

Identifying and estimating these double counted terms in gross exports in addition to

value-added exports have important implications for measuring each country's position in global

value-chains. For example, two countries can have identical ratios of value added exports to

gross exports but very different ratios of (a) and (b). Those countries that are mainly upstream in

global production chains, such as product design, tend to have a large value of (b) but a small

value of (a). In comparison, those countries mainly specializing in assembling imported

components to produce final products tend to have a small value of (b) but a big value of (a).

However, the existing literature lacks a uniform and transparent framework to compute exports

of value added, (a) and (b) simultaneously. Because one needs a gross exports accounting

(decomposition) framework to identify these conceptually different components from official

gross exports statistics, we venture to do this next. Without loss of generality, let us work with

country 1's gross exports first:

212221121221221

221211211112112

ybvybvxvVTybvybvxvVT

+=≡+=≡

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2121212 xaye += (10)

It says that country 1’s exports consist of final goods and intermediate goods. Combining (10)

with equation (8), we have

21221212112121121221211221212111

21221221211112212121112121221211112 ))((xabvxabvybvybvybvybv

xabvxabvybvybvxaybvbve+++++=

+++=++= (11)

A step by step proof of 1211212112122121212111 xabvybvybvxabv ++= can be found in appendix A.

Here we give the economic intuition behind it. The total value of country 1's intermediate

exports must include two types of value. First, it must include all value added by Country 1 in

its imports from Country 2. To see this, we note that in order for exported value produced by

Country 1 to come back through its imports, it must have first been embodied in Country 1’s

intermediate exports, which is 12112121121 xabvybv + . Second, it must include all value added

generated in Country 1 that is absorbed in Country 2 after being used as intermediate inputs by

Country 2, which is 22121 ybv .

Note that multiplying the Leontief inverse with intermediate goods exports leads to some

double counting of gross output and thus some value terms in exports. However, in order to

account for 100% of the value of country 1's intermediate goods exports and to identify what is

double counted, we have to include them into the accounting equation first. By decomposing the

last two terms further, we can see precisely what is double counted. Using the gross output

identity (equation (2)) 12111111 exayx ++= and 21222222 exayx ++= , it is easy to show that

121

11111

111 )1()1( eayax−− −+−= 21

12222

1222 )1()1( eayax

−− −+−= (12)

111

11)( yaI−− is the gross output needed to sustain final goods that are both produced and

consumed in country 1, using domestically produced intermediate goods; deducting it from

country 1's total gross output, what is left is the gross output needed to sustain country 1's

production of its gross exports e12. Therefore the two terms in right hand side of equation (12)

both have straightforward economic meanings. We can further show that

111

111111111

112112 )1()1( yaybyaab−− −−=− (see proof, also in Appendix A), which is the total gross

output needed to sustain final goods both produced and consumed in country 1, but using

intermediate goods that originated in Country 1 and shipped to Country 2 for processing before

being re-imported by Country 1 (gross output sold indirectly in the domestic market). These two

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parts plus 2112 yb sum to x11 in equation (5), which is the gross output of country 1 absorbed in

country 1 to sustain its domestic final demand both directly and indirectly. It indicates that

Country 1's domestic final demand is satisfied by three production channels: (1) 11111)( yaI −− is

part of country 1's gross output sold directly in the domestic market that is consumed there; (2)

2112 yb is part of country 1's gross output used as intermediate goods by country 2 produce final

goods that is consumed in country 1; (3) 111112112 )1( yaab −− is part of country 1's gross output used

as intermediate goods by country 2 to produced intermediate goods that is exported to country 1

to produce final goods in country 1 that is consumed there. They are all needed to sustain the

domestic final demand in country 1, but they differ in terms of how they participate in

international trade.

Replacing x1 by 121

11111

11 )1()1( eaya−− −+− and x2 by 21

12222

122 )1()1( eaya

−− −+− in

equation (11), and rearranging terms, we can fully decompose Country 1's gross exports into its

various value-added and double counted components as follows:

211

2212212221

221221212212

121

1121121111

112112121121

2212112111122121211112

)1(])1([)1(])1([

][

eaabvyaabvybveaabvyaabvybv

ybvybvebvebve

−−

−−

−+−++

−+−++

+=+=

.

(13)

While the algebra to arrive at equation (13) may be a bit tedious, expressing a country’s gross

exports as the sum of these eight terms on the right hand side of equation (13) is very useful. We

go over their economic interpretations systematically.

The first two terms in equation (13) (or the two terms in the first square bracket) are

value-added exports, i.e. country 1's domestic value-added absorbed outside country 1.

The third term, 21121 ybv is country 1's domestic value-added that is initially embodied in

its intermediate exports but is returned home as part of Country 1’s imports of the final good.

The fourth term, 111

1121121 )1( yaabv−− , is also Country 1’s domestic value added that is initially

exported by Country 1 as part of its intermediate goods to Country 2, but then is returned home

via its intermediate imports from country 2 to produce final goods that is absorbed at home. Both

the third and the fourth terms are domestic value added produced in Country 1, exported to

Country 2, but then return to and stay in Country 1. Both are counted at least twice in trade

statistics as they first leave Country 1 for Country 2, and then leave Country 2 for Country 1 (and

ultimately stay in Country 1). Note that the value represented by these two terms can be

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embodied in trade transactions that cross borders back and forth for more than twice as long as

they originate in Country 1and are ultimately consumed in Country 1.

The fifth term, 121

1121121 )1( eaabv−− , may be called a “pure double counted term.” The

reason for labeling it as such will become clear after we present a similar dissection of Country

2’s gross exports and a further decomposition of this term. This term only occurs when both

countries export intermediate goods. If at least one country does not export intermediate goods

(i.e., no two-way trade in intermediate goods), this terms disappears.

The sixth term, 12212 ybv , is the foreign value-added in country 1's gross exports of final

goods, and the seventh term, 221

2212212 )1( yaabv−− , is the foreign value-added in country 1's

gross exports of intermediate goods, they both ultimately go back to the foreign country and

consumed there.

The eighth (and the last) term is another pure double counted item in country 1's gross

exports. Similar to the 5th term, this term would disappear if at least one country does not export

intermediate goods.

In a similar way, we can express Country 2's gross exports as the sum of eight terms:

121

1121121111

112112121121

211

2212212221

221221212212

2122211212212222112121

)1(])1([)1(])1([

][

eaabvyaabvybveaabvyaabvybv

ybvybvebvebve

−−

−−

−+−++

−+−++

+=+=

(14)

Comparing equations (13) and (14), there are a few noteworthy features. First, the 3rd, 4th

and 5th terms in Country 1’s gross exports (13) are identical to the 6th, 7th, and 8th terms in

Country 2’s exports (14), and vice versa. This means that, the value added that is initially

produced and exported by Country 1 but then re-imported by Country 1, is exactly the same as

foreign value added in Country 2’s gross exports to Country 1. Symmetrically, the foreign value

added in Country 1’s gross exports to Country 2, is the same as Country 2’s value added, initially

produced and exported by Country 2, but re-appears as part of Country 1’s gross exports to

Country 2.

Second, while the 1st and 2nd terms in equations (13) and (14) constitute value-added

exports, all other terms are double counted components in a country's official exports statistics.

However, there are conceptually interesting differences among the 3rd and the 4th terms as the

first group, the 6th and the 7th terms as the second group, and the 5th and 8th terms as the third

group. The differences are revealed when comparing them to the two countries’ GDP. More

-15

111

111111

1121122112221212111

11112112221212111111

)1(]})1([{

)(

yavyaabybybybvybybybybvxvGDP

−− −+−+++=

+++==

precisely, a country’s GDP is the sum of its value-added exports plus its domestic value-added

consumed at home:

(15)

(16)

The last term in each GDP equation is value-added produced and consumed at home that are not

related to international trade; while the first four terms in the bracket in each GDP equation are

exactly the same as the first four terms in equations (13) and (14). It is easy to show that the sum

of global GDP always equals global final demand:

(17)

Equations (15) and (16) show that the 3rd and 4th terms in equations (13) and (14) are

counted as part of the home country’s GDP (even though they are not part of the home country’s

exports of value added). Because they represent a country’s domestic value-added that is initially

exported but imported back and consumed in the initial producing country, they are part of the

value-added created by domestic production factors. The 6th and 7th terms represent the foreign

value added in a Country’s exports that are ultimately absorbed in the foreign country. They are

counted once as part of the foreign country's GDP in equations (15) and (16). In comparison,

because a combination of the part of GDP that is consumed at home and exports of value added

yields 100% of a country’s GDP, the 5th and 8th terms are not part of either country’s GDP. In

this sense, they are “pure double counted terms.”

Subtracting global GDP from global gross exports using equations (13), (14), (15) and

(16) yields the following:6

(18)

Equation (18) shows clearly that besides the value added produced and consumed at

home (in the last square bracket), which is not a part of either country's gross exports, the 6th and

6 A step by step derivation is provided in Appendix A.

])1()1([)1(2)1(2])1([])1([

221

222111

111211

2212212121

1121121

221

22122112212111

11211221121212112

yavyaveaabveaabvyaabybvyaabybvGDPGDPee

−−−−

−−

−+−−−+−+

−++−+=−−+

2122212122121111

2221212111221121 )1()1(yyxaxaxxaxax

xaaxaaxvxvGDPGDP+=−−+−−=

−−+−−=+=+

221

222221

2212211221212211212

22221221212211212222

)1(]})1([{

)(

yavyaabybybybvybybybybvxvGDP

−− −+−+++=

+++==

-16

7th terms in equations (13), ])1([ 221

22122112212 yaabybv−−+ , and the 6th and 7th terms in equation

(14) ])1([ 111

11211221121 yaabybv−−+ , double counted only once as foreign value-added in the other

country's gross exports. Because the 3rd and 4th terms in (13) and (14) reflect part of the countries’

GDP, they are not double counted from the global GDP point of view. In comparison, both the

5th and 8th terms are over-counted twice relative to the global GDP since they are not a part of

either country’s GDP.

Third, the nature of the 5th and 8th terms can be understood further if we break them up

further. In particular, with a bit of algebra, they can be expressed as linear combinations of

components of the two countries’ final demand:

])1([)1( 111

11211212112122121121112112121

1121121 yaabvybvybvybvabeaabv−− −+++=− (19)

])1([)1( 221

22122121221221122112121221211

2212212 yaabvybvybvybvabeaabv−− −+++=− (20)7

The four terms inside the square bracket on the RHS of equation (19) are exactly the

same as the first four terms in the gross exports accounting equation (13). A similar statement

can be made about the four terms inside the square bracket in equation (20) to gross export

accounting equation (14). This means that the 5th and the 8th terms double counted a fraction of

both a country's value added exports and its domestic value added that has been initially exported

but are eventually returned home. Just to belabor the point, unlike the other terms in Equation

(13) that are parts of some countries’ GDP, the 5th and the 8th terms over-count the values that are

already captured by other terms in gross exports. Again, this feature suggests that they are “pure

double counted terms.”

Note, if Country 2 does not export any intermediate goods, then 21a = 21b =0, and the

entire RHS of equations (20) and (19) would vanish. Alternatively, if Country 1 does not export

intermediate goods, then 12a = 12b = 0, the entire RHS of equations (19) and (20) would also

vanish. In other words, the 5th and 8th terms exist only when two-way trade in intermediate goods

exist so that some value added is shipped back and forth as a part of intermediate trade between

the two countries. Because the eight components of equations (13) and (14) collectively

constitute 100% of a country's gross exports, missing any part, including the two pure double

counting terms, the accounting would not be complete.

7 Proofs of equations (19) and (20) are provided in Appendix A.

-17

Finally, while the 5th and 8th terms in Equation (13) are (double counted) values in

intermediate goods trade that are originated in Countries 1 and 2, respectively, we cannot directly

see where they are absorbed. By further partitioning the 5th and 8th terms, we can show where

they are finally absorbed and interpret the absorption by input/output economics. With a bit of

algebra, we can show8:

])1([])1([)1()1(

111

2112211221221

221221122112

211

2212212121

1121121

yaabybayaabybaeaabveaabv

−−

−−

−++−+=

−+− (21)

Therefore, the sum of 8th and 5th terms in equation (13) is equivalent to the sum of the four terms

in the RHS of equation (21) . From equation (5), it is easy to see that the terms in the square

bracket in equation (21) are parts of x22 and x11, which are trade-related portions of gross output

that are both originally produced and finally consumed in the source country; therefore they are

part of each country's gross intermediate exports (a12x2 and a21x1). Since the value added

embodied in those intermediate goods are already counted once in the production of each

country's GDP (the 3rd and 4th, terms in equations (13) and (14)), they are double counted in

value-added (GDP) terms. As we pointed earlier, the exact same terms will also appear in

Country 2's official exports statistics.

Due to the presence of these types of conceptually different double counting in a

country's gross exports, we may separately define “domestic value added in exports,”(“part of a

country’s GDP in its exports”) and “domestic content in exports.”The former excludes the pure

double counted intermediate exports that return home; whereas the latter is the former plus the

pure double counted term.

2.3 Using the accounting equation to generate measures of vertical specialization

We can relate the definitions of the three concepts to components in Equation (13).

Country 1’s exports of value added are the sum of the 1st and the 2nd term. It takes into account

both where the value is created and where it is absorbed. Country 1’s value added in its exports

is the sum of its exports of value added and the 3rd and the 4th terms. This concept takes into

account where the value is created but not where it is absorbed. Obviously, Country 1’s “value

added in its exports” is generally greater than its “exports of value added.” Finally, the domestic

content in Country 1’s exports is Country 1’s value added in its exports plus the 5th term, the 8 A step by step proof is in appendix A.

-18

double counted intermediate goods exports that are originated in Country 1. This last concept

also disregard where the value is ultimately absorbed. By assigning the 5th term to the domestic

content in Country 1’s exports, and the 8th term to the foreign content in Country 1’s exports, we

can achieve the property that the sum of the domestic content and the foreign content yields the

total gross exports. We will justify these definitions in what follows. We will also argue that one

of these measures may be more appropriate than others, depending on particular economic

applications.

We already show that the first two terms in Equation (13) correspond to value added

exports as proposed in Johnson and Noguera (2012). We now link other measures in the

literature to linear combinations of the components in the same gross exports accounting

equation. Following HIY’s original ideas, Koopman, Wang and Wei (2008 and 2012) have

shown that gross exports can be decomposed into domestic content and foreign content/vertical

specialization (VS) in a single country IO model without two-way international trade in

intermediate goods.

If we were to maintain HIY’s assumption that Country 1 does not export intermediate

good (i.e., 12a = 12b = 0), then the two pure double counted terms, or the 5th and the 8th terms in

equation (13) are zero. In this case, we can easily verify that the sum of the last four terms in

equation (13) is identical to the VS measure in the HIY (2001) paper.

To remove the restriction that HIY imposed on intermediate trade, we have to determine

how to allocate the two pure double counted terms. We choose to allocate the double counted

intermediate exports according to where they are originally produced. That is, even though the

5th term in equation (13) reflects pure double counting, it nonetheless is originally produced in

Country 1 and therefore can be treated as part of Country 1’s domestic content. Similarly, we

allocate the 8th term to the foreign content in Country 1’s exports. Such a definition is consistent

with HIY’s original idea that a country’s gross exports consist of either domestic or foreign

content and the major role of their VS measure is to quantify the extent to which intermediate

goods cross international boarder more than once. It is also computationally simple because the

share of domestic and foreign content can be obtained directly from the VB matrix.

However, since the 5th term reflects double counted intermediate goods in a country’s

gross exports, we may wish to exclude it if we are to consider which part of Country 1’s GDP is

exported. In particular, we define “domestic value added in Country 1’s gross exports”

-19

(regardless of where the exports are ultimately absorbed) as the sum of the first four terms on the

RHS of equation (13). This variable can be shown to equal to 121111 )1( eav −− , part of equation (12),

Country 1’s gross output identity.

])1([

])1([

}]1)1[()()1()1{()1(

)]()1[()1(

)]([)1(

)()1()1(

111

112112211212212112111

111

11111211212212112111

111111222112111211111

111

222221221121121221121

111

222212212122112112121

111

212121

111121

1111

yaabybvybvybvyabvybvybvybv

ybayybaybaavybybybaybaav

ybybybybayavxayaveavDV

−

−

−

−

−

−−

−+++=

−−+++=

−−++−+−−=

++++−=

++++−=

+−=−=

(22)

The derivation uses the property of inverse matrix 21121111 1)1( baba +=− and 22121211 )1( baba =− .

Following the same frame of thinking, we label only the sum of the 6th and 7th terms, a

subset of the foreign content, as the “foreign value added in Country 1’s exports,” and define

country 1’s VS as

12111211

2212212221

22122112212122121 )1()1(])1([ ebveaabvyaabybvebvVS −=−+−+==−−

(23)

The first two terms are foreign value-added or GDP in country 1's exports, and VS share of

country 1 equals 212bv . Therefore, such a measure of foreign content is a natural extension of

HIY's VS measure in a two-country world with unrestricted intermediate goods trade. Because

the VS share is defined this way, it is natural to define the domestic content share in country 1's

exports as 1- VS share or 111bv .

In a two-country world, Country 1’s VS1 is identical to its VS1* and Country 2’s VS:

121

1121121111

222112211212112111 )1(])1([*1VS1 eaabvyaabybvebvVS−− −+−+=== (24)

which is the sum of the 3rd, 4th, and 5th terms in equation (13). However, in a multi-country

setting to be discussed later, VS1* will only be a subset of VS1, and the latter will also include

some third country terms.

-20

2.4 Numerical examples

To enhance the intuition for our formula, we provide some numerical examples.

Example 1: One of the countries does not export intermediate goods, and does not have domestic

value added in its gross exports.

Country A has a gross output of 150, which is produced by combining domestic

intermediate goods of 50 and domestic value added of 100. It exports 70 units of its output to

Country B, consisting of 50 units of intermediate goods and 20 units of final goods. (Country A

also supplies 50 units at home as intermediate goods, and another 30 units at home as final

goods.)

Country B does something extremely simple: it produces a gross output of 50 units by

using 50 units of imported intermediate goods from Country A and adding no domestic value.

Country B then exports the entire 50 units of its output to Country A as final goods.

The two country’s production and trade relationship can be summarized by the following

inter-country input-output model:

=

++

+

=

50

1500502030

50150

00133.0

B

A

xx

,

In this extremely simple and therefore transparent example, it is clear the entire 50 units

of Country B’s gross exports consists of foreign value added (or imported intermediate goods

from Country A), and no domestic value added is exported since no value is added by Country

B’s production. So its VS share = 100%, and VAX ratio =0%.

Intuitively, out of the total of 70 units of Country A’s gross exports, 50 units return home

(as Country B’s exports to A). Therefore, Country A’s exports of value added that are ultimately

absorbed in Country B are 20 units. So, VS share =0%, and VAX ratio =20/70.

We can easily check that our gross exports accounting formula correctly back out these

terms. Since

=

00133.0

A

It is obvious v1= 0.67, and v2=0 and easy to work out that

=

105.15.1

B

=

0011

VB

-21

This indicates that country 1's share of domestic value-added in its exports is 100% and

country 2’s share is 0%.

++++

=

++++

=

=

005000307545

0502030

105.15.1

2222122121221121

2212121121121111

2221

1211

ybybybybybybybyb

xxxx

For Country A, the accounting equation indicates that the exports of domestic value

added are 0.67*30=20 units (20 for the first term, and zero for the second term). The domestic

value added that returns home as embedded in imported intermediates is 0.67*75=50 units

(consisting of 50 for the 3rd term, and zero for the 4th term). Our formula returns a value of zero

for the 6th and 7th terms, which correspond to zero foreign value added in Country A’s exports.

The two pure double counted terms take a value of zero, which is consistent with our theoretical

discussion that these terms will disappear if at least one of the countries does not export

intermediate goods.

For country B, our formula produces a zero value for exports of value added, and

indicates that 50 units (or 100%) of the country’s export come from foreign value added. All

other terms are zero. These are exactly consistent with our intuition.

To sum up, in this transparently simple example, we can work out a decomposition of a

country’s gross exports in our head, and understand the economic meaning of each of these terms.

We can verify easily that our gross exports accounting formula correctly generates these items.

Example 2: Two countries, one sector, and one of the countries does not export intermediate

good

Consider a world consisting of two countries (USA and CHN) and a single sector of

electronics. The two countries have identical gross exports and identical value added exports

(and hence identical VAX ratios). The point of this example is to show that the structure of the

“double counted” values in gross exports contains useful information.

Both USA and CHN have a gross output of 200. USA’s total output consists of 150 units

of intermediate goods (of which, 100 units are used at home and 50 units are exported) and 50

units of final goods (of which 30 are consumed at home and 20 are exported).

CHN’s total output consists of 50 units of intermediate goods (all used at home) and 150

units of final goods (of which, 70 units are exported and 80 units are used at home).

-22

By construction, both countries export 70 units of their output (50 units of intermediate

goods + 20 units of final goods exported by USA, and 70 units of final goods exported by CHN).

The domestic value added in USA’s output is therefore 100 (=value of gross output 200 –

value of domestic intermediate good of 100). Note no foreign value is used in USA’s production.

The domestic value added in CHN’s output is also 100 (= value of gross output 200 -value of

domestic intermediate goods of 50 – value of imported intermediate goods of 50). The input-

output relationship can be summarized as follows:

=

200200

2

1

xx

,

=

25.0025.05.0

A,

=

5.005.0

V

The two-country production and trade system can be written as an inter-country input output

(ICIO) model as follows

++

+

=

80702030

200200

25.0025.05.0

2

1

xx

,

The Leontief inverse matrix and the VB matrix can be computed easily as

=

=

33.1067.02

2221

1211

bbbb

B ,

=

67.0033.01

VB

We can break up each country’s gross output according to where it is ultimately absorbed

by rearrange each country's final demand as follows:

=

++++

=

=

7.1063.933.937.106

67.106033.9303.534069.4660

80702030

33.1067.02

2221

1211

xxxx

It is easy to verify that 12111 xxx += and 22212 xxx += . USA’s value-added export is 46.7

(v1x12=0.5*93.33), where 20 (=v1b11y12=0.5*2*20, first term in equation (13)) is the amount of

USA’s value added in its export of final goods that is absorbed in CHN, and 26.7(=v1b12y22

=0.5*0.667*80, second term in (13)) is the amount of USA’s value added in the export of

intermediate goods that is also absorbed in CHN. The amount of USA’s value added that is

embedded in its export of intermediate goods but returns home as part of CHN's export of final

goods is 23.3 (=v1b12y21=0.5*0.667*70, the 3rd term in (13)). Using the terminology of Daudin et

al, USA's VS1*=23.3, the same as the VS1* estimates obtained from our accounting equation

(13).

-23

By construction, foreign value added in USA’s exports equals zero (since CHN does not

export intermediate goods). It is easy to verify that the sum of USA’s exports of value added

(46.7) and the amount of returned value added (23.3) is 70, which is the value of its gross exports.

For CHN, its value added exports (that are absorbed in USA) are also 46.7 (=v2b22y21 =0.5*1.333*70, first term in equation (14)). It is entirely embedded in CHN’s exports of final

good to USA as CHN does not export intermediate good by assumption. However, CHN does

import intermediate goods from USA to produce both goods consumed in CHN and goods

exported to USA. The amount of USA’s value added used in the production of CHN’s exports is

23.3(= 21121 ybv =0.5*0.667*70, the 6th term in equation (14)). The sum of CHN’s exports of value

added (46.7) and the foreign value added in its exports (23.3) is 70, which is the same as CHN’s

gross exports.

In this example, both countries have identical gross exports and exports of value added,

and therefore identical VAX ratios. However, the reasons underlying why value added exports

deviate from the gross exports are different. For USA, the VAX ratio is less than one because

some of the value added that is initially exported returns home after being used as an

intermediate good by CHN in the latter’s production for exports. For CHN, the VAX ratio is less

than one because its production for exports uses intermediate goods from USA which embeds

USA’s value added.

In addition, the double counted items are also value added at some stage of production.

More precisely, the VS in CHN’s exports (23.3) is simultaneously a true value added from

USA’s viewpoint as it is value added by USA in its exports to CHN (that returned home)9, but a

“double counted item” from CHN’s viewpoint as it is not part of CHN’s value added.

Since we assume that CHN does not export intermediate goods (or a21=0 and b21 = 0),

there is no channel for CHN’s gross output to be used by USA in its production and for CHNs

gross output to be first exported and then returned home. Therefore, our gross exports accounting

equation produces the same estimate for VS as the HIY’s formula, i.e:

3.2370)25.01(25.0)1(3.2370667.05.0)1( 1121

112112212112 =×−×=−==××==−−− eaaebvbv

9 Because there is no foreign value-added in country 1's production, the 30 unit of domestic final demand are 100% its own value-added, just as its exports, so its GDP equal to 100. For country 2, the value-added in its exports and domestic final consumption also sum to 100.

-24

Domestic value-added share for CHN's exports equals (70-23.3)/70 = 0.667 = v2b22=0.667. There

is no difference between HIY's measure and our method in such a case since there are no pure

double counting terms due to two way trade in intermediate goods. Both VS in USA and VS1* in

CHN are zero. But this equality will not hold when we remove the assumption of a21=0 (which

will see in next example).

However, because there is domestic value-added embodied in USA's intermediate goods

exports that is eventually returned home, the domestic value-added share in USA's exports

equals to 1. As a result, using the VAX ratio (0.667) as a metric of the share of home country's

domestic value-added in its gross exports would produce an underestimate.

Example 3: Both countries export intermediate good in an inter country supply chain10

We now consider an example in which both countries export (and import) intermediate

goods in an inter-country supply chain. This example will show our accounting equation can

decompose a country's gross exports into various value-added and double counted components in

a way that is consistent with one’s intuition. We will also illustrate why and how our estimate of

VS1* in such a case differs from Daudin et al, why and how our estimate of the share of

domestic value-added (GDP) in exports differs from Johnson and Noguera's value-added to gross

exports ratio, and why and how our estimate of foreign value-added (GDP) in exports differs

from HIY's VS measure but our foreign content in exports generalizes it.

Suppose the world production and trade take place in five stages (in a year) as

summarized by Table 1. In Stage 1, perhaps a design stage, Country 1 uses labor to produce a

unit of Stage-1 output. This is exported to Country 2 as an input to Stage-2 production. In Stage

2, Country 2 adds a unit of labor to produce 2 units of Stage-2 output which are shipped back to

country 1 as an input to Stage-3 production. Country 1 adds another unit of labor to produce 3

units of Stage-3 output which are then exported to country 2 as an input to Stage-4 production.

In Stage 4, country 2 adds a unit of labor to produce 4 units of Stage-4 output which are shipped

back to country 1 as an input to Stage-5 production. The Stage-5 output is the final good. 3

units of the final good are exported to country 2, and 2 units are absorbed domestically in

country 1.

10 We are grateful to Peter Dixon and Maureen Rimmer for helping us to develop this instructive example.

-25

Suppose each unit of intermediate and final goods is worth $1. The total output in country

1 is $9, in country 2 is $6, the total value added (labor inputs) in the two countries is $3 and $2

respectively. The total exports from 1 to 2 and from 2 to 1 are $7 and $6, respectively; and the

exports of final goods from 1 to 2 and 2 to 1 are $3 and $0 respectively.

For this simple example of an international supply chain, we can decompose both

countries’ gross exports into value-added and double counted components by intuition without

using any equations. The intuitive decomposition is summarized in Table 2. We will then verify

that our exports decomposition formula produces exactly the same results.

We proceed as follows: Starting from the last stage (Stage 5), each country contributes $2

of value-added with their (previously produced) intermediate inputs, and Country 1 contributes

an additional $1 of labor input to produce a total of 5 units of the final good. We assume labor is

homogenous across countries. Since 2 units of the final good stay in Country 1 and 3 units are

consumed in Country 2, all the value-added embodied in intermediate inputs that are eventually

absorbed by each country should be split as 40% for country 1 and 60% for country 2, in

proportion to the units of the final good consumed by the two countries. Therefore, the total

value added exports from Country 1 to 2 are 0.6*$3=$1.8 (which is recorded in the cell in row

“total” and column 2a of Table 2). Similarly, Country 2’s exports of value added to 1 are

0.4*$2=$0.8 (which is recorded in the cell in row “total” and column 2b). Out of Country 1’s $7

of gross exports, the total amount of double counting, or the difference between its gross exports

and its value-added exports is $5.2 (=$7-$1.8). This is recorded in the cell in row “total” and

column 7a. Similarly, out of Country 2’s $6 of gross exports, the total amount of double

counting is $6-$0.8=$5.2, which is recorded in the cell in row “total” and column (7b).

The beauty of this simple example is that we can work out the structure of the double

counted values by intuition. Given what happens in Stage 5, we can split a country’s value

added in production in each of the earlier stages into the sum of value-added exports in that stage

(that is ultimately absorbed abroad) and the value added that is exported in that stage but returns

home next stage as part of its imports from the foreign country. Then the amount of exports in

each of the first 4 stages that are double counted can be computed as each stage's gross output

minus value added exports in that stage. In Stage 1, Country 1’s domestic value added is $1

(recorded in the cell (S1, 1a)). Since we know by Stage 5, 40% of the final good stays in Country

1, and 60% is exported to Country 2, we can split the $1 of domestic value added into $0.6 of

-26

Country 1’s exports of value added (recorded in the cell (S1, 2a)) and $0.4 of the domestic value

added that returns home in the next stage and eventually consumed at home in Stage 5 (recorded

in (S1, 3a)). Out of Country 1’s gross exports of $1 in Stage 1, the total double counted amount

is the difference between its gross exports and value added exports, or $1-$0.6=$0.4, as recorded

in (S1, 7a).

In Stage 2, Country 2 uses $1 of intermediate good from Country 1 as an input together

with its additional $1 of labor to produce $2 exports. Its domestic value added is $1 (recorded in

(S2, 1b)). Again, since we know the split of the final good consumption in the two countries in

Stage 5, we can split Country 2’s domestic value added into $0.4 of its exports of value added

(recorded in (S2, 2b)) and $0.6 of domestic value added that will return home in Stage 3 and

eventually consumed at home in Stage 5(recorded in S2, 4b)). Recall that out of $1 of

intermediate good that Country 2 imports from Country 1, $0.4 will go back to Country 1 and be

consumed there eventually. This is recorded in (S2, 5b), which is numerically identical to (S1,

3a). The remaining $0.6 is double counted intermediate goods, and is recorded in (S2, 6b). This

can also be verified in the following way. Since we know Country 2’s gross exports in Stage 2 is

$2 but its value added exports are only $0.4, the total amount of double counting in this stage’s

gross exports must be the difference between the two, or $1.6 as recorded in (S2, 7b).Therefore,

the “pure double counted” portion of foreign intermediate good has to equal $1.6 (S2, 7b) -$0.6

(S2, 3b) - $0.4 (S2, 5b), which equals to $0.6, as recorded in (S2, 6b). This amount represents the

part of Country 1’s Stage 1 intermediate good exports that cross borders more than twice before

it can be embed in the final goods for consumption.

In Stage 3, Country 1 uses $2 of imported intermediate goods from Country 2 as an input

with its additional $1 of labor to produce $3 exports. Country 1’s domestic value added is $1 (S3,

1a). Again, because 60% of the final good will be eventually absorbed in the foreign country, the

$1 of domestic value added can be split into $0.6 of Country 1’s exports of value added (S3, 2a)

and $0.4 of the domestic value added that is exported in Stage 3 but will return in Stage 4 and

eventually consumed there in Stage 5(S3, 3a). Furthermore, the Stage 3 production does use

imported intermediate good from the previous stage. The amount of foreign value added

embedded in its intermediate good imported from Country 2 that is not pure double counting

should be the same as Country 2’s domestic value added that is sent to Country 1 in Stage 2 but

returns home and will be eventually absorbed there. We know that amount is $0.6 (S2, 3b).

-27

Therefore, the amount of foreign value added that is used in Country 1’s Stage 3 production for

exports and that will be eventually absorbed in Country 2 should be the same as $0.6 in (S3, 5a).

Because the value of Country 1’s stage 1 exports ($1) is already counted three times by

the time Stage 3 exports take place, we record that amount as a pure double counting item in (S3,

4a). Since we know out of $3 of Country 1’s gross exports in Stage 3, only $0.6 is exports of

value added that will eventually be absorbed abroad, $3-$0.6=$2.4 represents the total amount of

double counting in this stage’s gross exports, and is recorded in (S3, 7a). Out of the $1 foreign

value added from Stage 2, since the amount that will go back to the foreign country and is

absorbed there is 0.6 (S3, 5a), the amount of pure double counting must be $1-$0.6=$0.4, as

recorded in (S3, 6a).

One way to check the sensibility of our reasoning is to compare the total amount of

double counting in Stage-3 gross exports with the sum of the double counted components. Out of

Country 1’s $3 of gross exports in Stage 3, we know the total amount of double counting is $2.4

(recorded in (S3, 7a)). We can check that the sum of the double counted components in Country

1’s exports in this stage (the sum of (S3, 3a), (S3, 4a), (S3, 5a), and (S3, 6a)) is also $2.4.

We now move to Stage 4, when Country 2 combines $1 of domestic value (recorded in

(S4, 1b)) with $3 of intermediate goods imported from Country 1 in the previous stage, and

exports $4 of intermediate goods in gross terms to Country 1. Given that 40% of the final good

will be absorbed in Country 1 by stage 5, we can split Country 2’s $1 domestic value added in

this stage into $0.4 which is Country 2’s value added exports (S4, 2b), and $0.6 which is the

amount of its domestic value added that will return home in Stage 5 and be absorbed at home (S4,

3b). Country 2’s gross exports in this stage also contain 40% of County 1’s value added from the

previous stage, recorded as $0.4 in (S4, 5b).

By symmetry, the pure double counting amount in (S4, 4b) must be the same as (S3, 4a),

which is $1. Let us next work out the pure double counting term in (S4, 6b). First, out of Country

2’s $4 gross exports in Stage 4, only $0.4 is value added exports, we know the total amount of

double counting must be $3.6, which is recorded in (S4, 7b). Second, we also know $3.6 of the

total amount of double counting must be equal to the sum of the double counted components, or

the sum of (S4, 3b), (S4, 4b), (S4, 5b) and (S4, 6b). This implies that (S4, 6b) should be $1.6.

The economic meaning of (S4, 6b) is repeated double counting of the intermediate goods that

have been double counted in previous rounds of trade.

-28

We now go to Stage 5. Because this is the final stage in which the final good is produced

by Country 1 but distributed 40% and 60% in Countries 1 and 2, respectively, we record the

values somewhat differently from the earlier stages (when the entire production was exported).

While Country 1’s domestic value added in the production is $1 in this stage, only 60% of the

final good is exported. So we record the amount of domestic value-added in Country 1’s exports

as $0.6 (S5, 1a). The amount of Country 1’s value added exports (that is absorbed in Country 2)

is also $0.6, as recorded in (S5, 2a).

Since Stage 5 production uses imported intermediate good from the previous stage, it

embeds foreign value added from Stage 4. The amount of foreign value added from Stage 4 that

is used in Country 1’s Stage 5 production and eventually absorbed in the foreign country is

proportional to the amount of the final good that is exported from Country 1 to 2. This means (S5,

5a) is $0.6. This of course is the same value as in (S4, 3b).

To determine the value in (S5, 4a), we note that the total value added from Country 1 in

the first and the 3rd stages are $1. Both values are counted as part of Country 2’s intermediate

exports in Stage 4. Since only 60% of the final good are exported, the pure double counting

associated with the domestically produced intermediate goods in the previous stages is $2*0.6 =

$1.2.

To determine the value of (S5, 6a), we first note that the total amount of double counting

in Stage 5 exports is the difference between the value of gross exports in that stage ($3) and the

value added exports in that stage ($0.6), which is $2.4, as recorded in (S5, 7a). The value in (S5,

6a) would simply be the difference between $2.4 and the sum of the values in (S5, 2a), (S5, 4a),

and (S5, 5a), which yields $0.6. The amount in (S5, 6a) represents the value that is originally

created in Country 2 but has been counted multiple times beyond the value added of Country 2

already assigned to Countries 1 and 2.

We can check the sensibility of the discussion by summing over the values across the five

stages. For example, when we sum up the values over all stages in Column (2a), we obtain 1.8,

which is exactly the amount of Country 1’s value added exports that we intuitively think should

be. Summing up the values in Column (7a) across the five stages yields $5.2, which is the same

as what we obtain intuitively earlier.

Separately, we can apply our decomposition formula and generate the measurements of

the same set of economic concepts. To do so, we note that the five stages in this example are best

-29

represented by 5 sectors (e.g., car windows, paint on a car, rubber tires on a car and a whole car

are considered in separate sectors). Applying a multi-sector version of our gross exports

accounting equations (13) and (14), we obtain estimates of the various components of the gross

trade and summarize them in Table 3. (The computation details can be found in Appendix C.) It

can be checked easily that the numbers in Table 3 generated by our formula match exactly with

the corresponding ones that one can intuitively work out in Table 2. In particular, Country 1’s

value added exports (that are absorbed abroad) from our formula in Table 3 are $1.8, exactly as

that in Table 2. In comparison, the total domestic value added in Country 1’s exports (that does

not exclude exported value added that returns home but does exclude the pure double counted

term) is $2.6. This example confirms our theoretical discussion that value-added exports are

generally smaller than domestic value-added (GDP) in exports and domestic content in exports.

If one is interested in the share of domestic value added in a country’s exports, then the VAX

ratio is not the right metric.

From Table 3, the VS measure produced by our decomposition formula (13) is 2.2. The

intuitive discussion in connection with Table 2 illustrates why we argue that the VS measure is

not a 'net' concept and is not equal to foreign value added in a country’s gross exports. The

fundamental reason is that the VS measure has to include some pure double counted terms.

(Again, these pure double counting terms would disappear if we use the HIY assumption that at

least one of the countries does not export intermediate good.)

The more intermediate trade crosses border, the larger these double counted foreign

intermediates imports are. With two-way intermediate trade, the part of foreign GDP that is

embodied in the home country's gross exports will always be smaller than the VS measure.

Relative to the original VS measure, our generalized measure includes double counted

intermediate exports produced by the foreign country that may cross border several times (v8).

The numerical results also show HIY's convention that a country's gross exports is equal to

domestic content plus vertical specialization is also maintained by our accounting equation (as

long as one defines domestic content and vertical specialization appropriately).

Finally, this example also shows that if one only considers returning domestic value-

added in final goods, while excluding domestic content returning home via intermediate goods

imports, such as Daudin et al (2011), then one would under-estimate VS1*. In this example, if

one applies Daudin et al’s narrow definition of VS1*, it would be zero as indicated by v3 in

-30

Table 3. If one also includes returning domestic value added in intermediate good and a pure

double counting term, VS1* would become $3 instead. Our redefined measure of VS1* is more

complete.

2.5 The General Case of G Countries and N Sectors

We now discuss the general case with any arbitrary number of countries and sectors. The

ICIO model, gross output decomposition matrix, value-added by source shares matrix, are given

succinctly by block matrix notations:

(25)

(26)

=

GGGGGGG

G

G

BVBVBV

BVBVBVBVBVBV

VB

21

22222212

11121111

(27)

With G countries and N sectors, A, and B are GN×GN matrices; V and VB are G×GN matrices.

Vs denotes a 1 by N row vector of direct value-added coefficient, Asr is a N×N block input-output

coefficient matrix, Bsr denotes the N×N block Leontief inverse matrix, which is the total

requirement matrix that gives the amount of gross output in producing country s required for a

one-unit increase in final demand in destination country r. Xsr is a N×1 gross output vector give

gross output produced in s and absorbed in r. Xs = ∑G

rsrX is also a N×1 vector that gives country

s' total gross output. Ysr is a N×1 vector give final goods produced in s and consumed in r. Ys =

=

−−−

−−−−−−

=

∑

∑

∑−

GGGGG

G

G

G

rGr

G

rr

G

rr

GGGG

G

G

G Y

YY

BBB

BBBBBB

Y

Y

Y

AIAA

AAIAAAAI

X

XX

2

1

21

22221

11211

2

11

21

22221

11211

2

1

=

GGGG

G

G

GGGG

G

G

GGGG

G

G

YYY

YYYYYY

BBB

BBBBBB

XXX

XXXXXX

21

22221

11211

21

22221

11211

21

22221

11211

-31

∑G

rsrY is also a N×1 vector that gives the global use of s’ final goods. Both the gross output

decomposition and final demand matrix in equation (26) are GN×G matrices. Let sV̂ be a N by N diagonal matrix with direct value-added coefficients along the

diagonal. (Note sV̂ has a dimension that is different from Vs). We can define a GN by GN

diagonal value-added coefficient matrix as

=

∧

∧

∧

∧

GV

V

V

V

00

00

00

2

1

(28)

Using the similar intuition as we used to derive equation (9) in the two country one sector

case, we can obtain domestic value-added in a country's gross output by multiplying this value-

added coefficient matrix with the right hand side of equation (26), the gross output

decomposition matrix. This will result in a GN by G value-added production matrix ∧

VBY as

=

=

∑∑∑

∑∑∑

∑∑∑

∧

∧

∧

G

rrGGrG

G

rrGrG

G

rrGrG

G

rrGr

G

rrr

G

rrr

G

rrGr

G

rrr

G

rrr

GGGG

G

G

G YBVYBVYBV

YBVYBVYBV

YBVYBVYBV

XXX

XXXXXX

V

V

V

BYV

21

22222122

11211111

21

22221

11211

2

1

00

00

00

ˆ (29)

Elements in the diagonal columns give each country's production of value-added absorbed at

home. As in the two country case, exports of value-added can be defined as the elements in the

off-diagonal columns of this GN by G matrix as

(30)

Obviously, it excludes value-added produced by the home country that returns home after

being processed abroad. A country's total value-added exports to the world equal:

(31)

∑=≡G

ggrsgssrssr YBVXVVT

∑∑∑≠ =≠

==G

sr

G

ggrsgs

G

srsrs YBVVXVT

1*

-32

By rewriting equation (31) into three groups according to where and how the value-added

exports are absorbed, we obtain decomposition as follows:

∑∑∑∑≠ ≠≠≠

++=G

srrt

G

rstsrsrr

G

srsrssr

G

srssss YBVYBVYBVVT

,* (32)

11

This is the value-added export decomposition equation in terms of all countries’ final demands.

The first term is value-added in the country's final goods exports; the second term is value-added

in the country's intermediate exports used by the direct importer to produce final goods

consumed by the direct importer; the third term is value-added in the country's intermediate

exports used by the direct importing country to produce final goods for third countries.

Comparing with equation (9), we can see clearly what is missing in our two country case: it is

the re-export of value-added via third countries, the last term of the RHS of equation (32),

because the distinction between value-added exports from direct and indirect sources only can be

made in a three or more country setting.

Define a country’s gross exports to the world as:

∑∑≠≠

+==G

srsrrsr

G

srsrs YXAEE )(* (33)

Using the logic similar to the derivation of equations (11), we can first decompose a country's

gross exports to its various components as follows:

∑∑∑∑∑ ∑

∑

≠ ≠≠ ≠≠ ≠

≠

++++=

+=

G

st

G

srrsrtst

G

st

G

srsrtst

G

sr

G

srsrssrsrssrss

s

G

srrsrsssss

XABVYBVXABVYBVVT

EBVEBVuE

}{}{*

***

(34)12

Based on the gross output identity for each country *sssssss EXAYX ++= , we have,

*11 )()( ssssssss EAIYAIX

−− −+−= *11 )()( rrrrrrrr EAIYAIX

−− −+−= (35)

Replace Xs and Xr in equation (34), insert equation (32), we obtain the G country, N sector

generalized version of gross exports accounting equation as follows:

11 This value-added exports decomposition also could be done at the bilateral level, however, it is different from equation (15) in Johnson and Noguera (2012). They split bilateral gross exports to three groups. 12 The step by step proof is provided in online Appendix B.

-33

(36)